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On the optimality of training signals for MMSE channel estimation in MIMOOFDM systems
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 105 (2015)
Abstract
In this paper, we investigate the optimality of training signals for linear minimum mean square error (LMMSE) channel estimation in multipleinput multipleoutput (MIMO) orthogonal frequency division multiplexing (OFDM) with frequencyselective fading channels. This is a very challenging problem due to its mathematical intractability and has not been analytically solved in the literature. Using the Lagrange multiplier method, we derive the optimality conditions for training signal design. Important findings revealed on optimal training signals are twofold: (i) the energies of the training signals on each subcarrier are equal, and (ii) on each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy. We verify that our results are in line with the design principles that have been derived in singlecarrier MIMO systems. Two types of optimal training signal examples that satisfy the optimality conditions are presented for practical implementations in MIMOOFDM systems. Simulation results show that the training signals based on the optimality conditions outperform other nonoptimal training signals in terms of channel estimation performance.
Introduction
Recently, increasing interest has been concentrated on multipleinput multipleoutput (MIMO) orthogonal frequency division multiplexing (OFDM) for broadband wireless communication. The combination of OFDM with MIMO exploits the benefits from both techniques, i.e., the robustness to combat multipath delay spread and an increase in system capacity [15]. For a practical implementation of MIMOOFDM systems, channel estimation becomes very important for the system performance. Imperfect channel estimation typically leads to the increase of error rates and reduces transmission efficiency. In this study, we focus on the problem of designing MIMOOFDM training signals for channel estimation, a critical component in many modern wireless communication systems.
Several approaches on optimal training signal design have been proposed in the literature. In the absence of prior statistical information about the channel, simple leastsquares channel estimation is used. In [6], the optimal placement of training signals is studied for a singleinput singleoutput (SISO) OFDM case. The optimal training design is extended to the MIMOOFDM case in [7]. In [8], more general training structures for MIMOOFDM are presented that utilize frequency division, time division, and code division multiplexing.
In the presence of prior statistical information about the channel, a more efficient channel estimation technique can be used. Linear minimum mean square error (LMMSE) estimators that incorporate prior knowledge to improve channel estimation are known to be optimal for this case. In [9,10], optimal conditions for training signals are studied for SISO frequencyselective fading channels. In [11], the results are extended to MIMO frequencyselective fading channels, and it is revealed that training signals across transmit antennas should be orthogonal and training signals should be equipowered. The results are very simple but effective and thus have been widely used as design guidelines for recent wireless systems. One can intuitively generalize the main principles to multicarrier systems while the optimality of the principles has remained unproved for the multicarrier systems, i.e., OFDM. For example, downlink reference signals in commercial LTE systems are designed with the same principles. In [12], an optimal training design for both leastsquares estimators and LMMSE estimators is studied assuming cyclic delay diversity OFDM systems.
Meanwhile, research interests have been shifted to more practical issues on MIMO channel estimations. In [13,14], a training signal design in the existence of inphase and quadrature imbalances is considered for SISOOFDM and MIMOOFDM systems. A robust training signal design for LMMSE channel estimator in case of imperfect knowledge of secondorder characteristics of channels is studied in [1517]. Efficient algorithms that exploit the spatial correlation of MIMO channels are proposed for training signal design in [18,19].
In this paper, we aim to complete the puzzle with the missing piece. We directly tackle a multicarrier system model and solve the optimality conditions of training signals for LMMSE estimator in MIMOOFDM systems. This is quite challenging because simultaneous considerations on all MIMO dimensions, channel statistics, multicarriers, and multisymbols lead to an extremely complex modeling and mathematically intractable problem. This has never been solved in the literature to the best of our knowledge. We analytically derive the optimality conditions using the Lagrange multiplier method, which is the principle contribution of this study.
The remainder of the paper is organized as follows. Section 2 describes the system model of our work. After Section 3 analytically derives the optimality conditions, the optimal training signal design is discussed in Section 4. Section 5 presents two types of optimal training signal examples for practical MIMOOFDM systems. Simulation results are provided in Section 6. Finally, Section 7 draws conclusions.
Notations: Uppercase and lowercase boldface letters are used for matrices and vectors, respectively. The superscript ‘ ∗’ denotes the conjugate transpose, superscript ‘T’ denotes the transpose, and superscript ‘ −1’ denotes the matrix/vector inverse. We will use \(\mathbb {E} \left [ \cdot \right ] \) for expectation, v e c(·) for matrix vectorization, t r[·] for the matrix trace, ⊗ for the Kronecker product, and I _{ N } for the N×N identity matrix.
System model
We consider a MIMOOFDM system with Q transmit and P receive antennas and perform an analysis in the frequency domain to search for the properties of an optimal training signal. For the best analytical tractability, we will work directly in the frequency domain. The number of OFDM subcarriers is N, and we consider a block of M OFDM symbols transmitted across the channel. We use the notation x _{ n,q,m } to denote the training signal transmitted on the qth transmit antenna in the mth symbol and on the nth subcarrier. For the nth subcarrier, we may, therefore, consider that we transmit a matrix of training signals, X _{ n }, where x _{ n,q,m } is the element in the qth row and mth column. The following notations are used in this paper:

X _{ n }: Training signal matrix for the nth subcarrier (Q×M),

Y _{ n }: Received signal matrix for the nth subcarrier (P×M),

H _{ n }: MIMO channel matrix for the nth subcarrier (P×Q),

W _{ n }: Received noise matrix for the nth subcarrier (P×M).
Considering frequencyselective fading, a signal model is given as
where all elements of H _{ n } are uncorrelated, and all noise variables are independent, i.e., \(\mathbb {E} \left [ \mathbf {W}_{n} \mathbf {W}_{n}^{*} \right ] = \sigma ^{2} \mathbf {I}\). Aggregating the matrices for all subcarriers gives
where x,y,h,w are respectively M N Q×1,M N P×1,N P Q×1, and M N P×1 matrices. By using a wellknown result regarding the v e c operation, i.e., v e c(A X B)=(B ^{T}⊗A)v e c(X), (1) becomes
or can be rewritten as
Combining all the N equations in (3) yields
or shortly denoted as
where X (M N P×N P Q) is defined as
Analysis on optimality conditions
In this section, the problem formulation for minimizing the LMMSE channel estimation errors in MIMOOFDM systems is presented first. Then, optimal conditions for the training signals are derived for an optimal training signal design.
Problem formulation
The standard LMMSE estimate of h, based upon the observation of y, becomes
By using (4), this becomes
The covariance matrix of the error can be expressed as
Using the wellknown matrix identity,
(8) can be rewritten as
Our goal is to find the matrix X in the form of (5) that minimizes (10), subject to the total transmit energy constraint.
Based on the assumptions of the channel, it can be observed that
where R (N×N) is a Toeplitz Hermitian matrix of the form
We also have
where Z (M N×N Q) is defined as
Furthermore,
Using some basic properties regarding ⊗ operations and expressions in (11) ∼(15), the MMSE in (10) can be rewritten as
In addition, it is known that, for any matrices A and B, there exists a permutation matrix Π such that A⊗B=Π(B⊗A)Π ^{T} [22]. Let Π be a permutation matrix such that R⊗I _{ Q }=Π(I _{ Q }⊗R)Π ^{T}. Using this relation in (16) yields
Note that row permutations by Π followed by column permutations by the same Π does not change the trace of the matrix because any diagonal parameter may be permuted but still remains in the diagonal position. Thus, we have
Finally, our optimization problem is formulated as a constrained optimization problem as
where E _{total} is the total transmit energy.
Optimality conditions
For simplicity of notations, we define D (N Q×N Q) and A (N Q×N Q) as
and
where Z ^{∗} Z is Hermitian and block diagonal. We solve the constrained optimization problem using the Lagrange multiplier method. By letting μ be the Lagrange multiplier, the Lagrangian is expressed as
To obtain the optimal solution, we set the derivatives of the Lagrangian function J(D,μ) to zeros as
where x is an arbitrary parameter in D. We split our approaches into two cases: diagonal parameters and offdiagonal parameters in matrix D. To find the required derivatives, we will use the following property. For any matrix A depending on a parameter x [22],
which directly follows from the fact that A A ^{−1}=I.
We first consider the diagonal parameters x in D and derive the following lemma.
Lemma 1.
By denoting the kth column of A ^{−1} as a _{ k }, the following condition should be satisfied in order to achieve \(\frac {\partial }{\partial x} J \left (\mathbf {D}, \mu \right) = 0\).
where c is a constant.
Proof.
Let x be the kth diagonal parameter in D. We define the derivative of A with respect to x as
where E is a diagonal matrix in which only the kth diagonal element has a value of one. Using associating (24) with (22), we have
where we use the fact that A ^{−1} is a Hermitian matrix in the second line. Applying \(\frac {\partial }{\partial x} J \left (\mathbf {D}, \mu \right) = 0\) finally gives
which holds for all diagonal elements in D without loss of generality. This completes the proof.
We then consider the offdiagonal parameters χ in D and derive the following lemma. Note that offdiagonal parameters are complex numbers. We split each parameter into two real parameters as χ=x+i y.
Lemma 2.
By denoting the kth column of A ^{−1} as a _{ k }, the following condition should be satisfied in order to achieve \(\frac {\partial }{\partial x} J \left (\mathbf {D}, \mu \right) = 0\).
Proof.
Let χ appear in the kth column and in the lth row in D. We first focus on the real part x. We define the derivative of A with respect to x as \(\frac {\partial \mathbf {A}}{\partial x} = \mathbf {E}\), where E is defined as a sparse matrix in which only the element in the kth column lth row and the element in the lth column kth row have a value of one. Using associating (24) with (22), we have
where we use the fact that A ^{−1} is a Hermitian matrix in the second line. Applying \(\frac {\partial }{\partial x} J \left (\mathbf {D}, \mu \right) = 0\) finally gives
where Re[·] denotes the real part of a complex number. Then, similar derivations with respect to the imaginary part y yield
where Im[·] denotes the imaginary part of a complex number. Combining both results provides
which holds for all offdiagonal elements in D without loss of generality. This completes the proof.
Optimal training signal design
In this section, we provide an optimal training signal satisfying the optimality conditions derived in the previous section.
Suppose that the matrix D is a multiple of the identity, i.e., D=c I, where c is a real constant. Because R is a Toeplitz Hermitian matrix, it can be written as
where F and Λ are a unitary matrix and a diagonal matrix, respectively. In addition, we may write
For shorter notations,
where \(\bar {\mathbf {F}}\) and \(\bar {\mathbf {\Lambda }}^{1}\) are respectively defined as
Associating (37) in (21) gives
where C is a Toeplitz Hermitian matrix. Thus, it is clear that D=c I satisfies both the optimality conditions C1 in (25) and C2 in (29). Solving D=Π ^{T} Z ^{∗} Z Π=c I for Z ^{∗} Z yields
From (14) and (40), we finally reach the following theorem on the optimal training signal.
Theorem (Optimal training signal).
In MIMOOFDM systems where a sequence of a training signal is transmitted at the transmitter through Q transmit antennas, N subcarriers, M OFDM symbols with the total transmit power of E _{total}, and LMMSE channel estimation is performed at the receiver upon the reception of M OFDM symbols from P received antennas, the training signal is ‘optimal’ in terms of minimizing channel estimation errors if the training signal satisfies the following conditions:

The energy of the training signal in each subcarrier is equal, i.e.,
$$\begin{array}{*{20}l} \mathbf{tr} \left[ \mathbf{X}_{n} \mathbf{X}_{n}^{*} \right] = \frac{E_{\text{total}}}{N}, \quad \text{for} \quad n = 1,2, \ldots, N. \end{array} $$((41)) 
On each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy, i.e.,
$$\begin{array}{*{20}l} \mathbf{X}_{n} \mathbf{X}_{n}^{*} = \frac{E_{\text{total}}}{NQ} \mathbf{I}, \quad \text{for} \quad n = 1,2, \ldots, N. \end{array} $$((42))
Examples of optimal training signals
In this section, we use the theorem revealed in the previous section as a design guideline and present two examples of optimal training signal implementations. These designs are practical owing to its simple structure. Note that optimal designs of training signals are not limited to the following cases.
Sequential transmission on antennas
Assume that the number of OFDM symbols transmitted is equal to the number of transmit antennas, M=Q, and let
where \(z_{i}^{(n)}\) are arbitrary complex numbers but satisfying \(\left  z_{i}^{(n)} \right ^{2} = \frac {E_{\text {total}}}{NQ}\). This implementation implies that each successive OFDM symbol is transmitted on a different antenna in a roundrobin fashion. Figure 1 illustrates an example of the optimal training signal design in this type of optimal training signal implementation.
Interlaced transmission on antennas
Assume that the number of OFDM symbols transmitted is equal to the number of transmit antennas, M=Q, and N is a multiple of Q. In this implementation, we transmit simultaneously from every antenna in each symbol interval, but each antenna uses only every Qth subcarrier. Let matrix Φ represent a cyclic shift operation, which causes a cyclic shift by one element in the upward direction. For example, if Q=3, we have
and it operates as
We define the training signal matrix by
where Φ ^{n} is the nth power of Φ,
and z _{ i } are arbitrary complex numbers but satisfying \(\left  z_{i}^{(n)} \right ^{2} = \frac {E_{\text {total}}}{NQ}\). This implementation implies that each antenna always transmit in every OFDM symbol but using only every Qth subcarrier. During the first symbol, for example, Antenna 1 uses subcarriers 1,Q+1,2Q+1,…, Antenna Q uses subcarriers 2,Q+2,2Q+2,…, etc. During the second symbol, Antenna 1 uses subcarriers 2,Q+2,2Q+2,…, and Antenna Q uses subcarriers 3,Q+3,2Q+3,…, and so on. Figure 2 illustrates an example of the optimal training signal design in this type of optimal training signal implementation.
Simulation results
In this section, we verify the optimality of the training signals through extensive computer simulations. We consider a MIMOOFDM system where the number of TX antennas is Q=4, the number of RX antennas is P=4, the number of OFDM subcarriers is N=128, and the length of the cyclic prefix is 32. The number of OFDM symbols for a training signal is set to M=4. A wide sense stationary uncorrelated scattering (WSSUS) model is considered for a multipath channel [23]. A multipath intensity profile of an exponential distribution is used where the number of delay taps is L=128 and an exponentially decaying factor is α. Doppler frequency is assumed to be zero. The optimal training signal shown in Figure 1 is used in the simulations. Five nonoptimal training signals are also generated for performance comparisons. Unlike the optimal training signal that satisfies the conditions in (41) and (42), the nonoptimal training signals are created at random but they all satisfy the total transmit power E _{total}.
Figure 3 compares the LMMSE channel estimation performances of different training signals. As the SNR increases, the channel estimation error decreases. The optimal training signal shows a considerable performance gap compared to the other nonoptimal training signals. For example, a SNR gain using the optimal training signal is more than 5 dB in most cases. Figure 4 shows how the LMMSE channel estimation performance varies with the multipath intensity profile. With a small exponentially decaying factor, large number of multipaths become dominant, resulting in high frequency selectivity and relatively low LMMSE channel estimation performance. As a small exponentially decaying factor increases, the number of dominant multipaths decreases, resulting in low frequency selectivity and relatively high LMMSE channel estimation performance. Again, the optimal training signal provides a huge performance gap compared to the other nonoptimal training signals in all cases. Figure 5 shows the LMMSE channel estimation performance with respect to the antenna dimension. As the number of antennas increases, the MMSE of the optimal training signal does not increase while those of nonoptimal training signals continuously increases. Accordingly, the performance gap between the optimal and nonoptimal training signals also increase. This is because the optimal training signal is designed such that training signals transmitted from the different antennas are orthogonal. In nonoptimal training signal, training signals simultaneously transmitted from different antennas collide and interfere each other, which degrades the LMMSE channel estimation performance.
Conclusions
In this paper, optimality conditions are analytically derived and design guidelines for the optimal training signals are provided for LMMSE channel estimation for MIMOOFDM. On the basis of the analysis, we clearly reveal that the training signal that satisfies the following is optimal: (i) the energy of the training signal on each subcarrier is equal, and (ii) on each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy. Interestingly, the optimality conditions of training signals for LMMSE estimator in MIMOOFDM systems are basically in line with design principles known from singlecarrier MIMO systems; training signals across transmit antennas should be orthogonal and training signals should be equipowered. We mathematically prove that the simple generalization of the design principles with an additional dimension, i.e., multicarriers, still holds optimality. This work is important because the optimality conditions of training signals for LMMSE estimation in MIMOOFDM systems have been mathematically proved. Future research may include an extension of the results to more practical channel statistics, e.g., correlated channels and timevarying channels.
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Jo, J., Sohn, I. On the optimality of training signals for MMSE channel estimation in MIMOOFDM systems. J Wireless Com Network 2015, 105 (2015). https://doi.org/10.1186/s136380150345y
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Keywords
 Optimal training signal
 MMSE channel estimation
 MIMO
 OFDM
 Frequencyselective fading